Two secant segments intersecting at the point in the exterior of the circle

When two secant lines intersect each other outside a circle, the products of their segments are equal.

(Note: Each segment is measured from the outside point)

Try this In the figure below, drag the orange dots around to reposition the secant lines. You can see from the calculations that the two products are always the same. (Note: Because the lengths are rounded to one decimal place for clarity, the calculations may come out slightly differently on your calculator.)

See also Intersecting Secant Angles Theorem.

This theorem works like this: If you have a point outside a circle and draw two secant lines (PAB, PCD) from it, there is a relationship between the line segments formed. Refer to the figure above. If you multiply the length of PA by the length of PB, you will get the same result as when you do the same thing to the other secant line.

More formally: When two secant lines AB and CD intersect outside the circle at a point P, then

PA.PB = PC.PD

It is important to get the line segments right. The four segments we are talking about here all start at P, and some overlap each other along part of their length; PA overlaps PB, and PC overlaps PD.

Relationship to Tangent-Secant Theorem

In the figure above, drag point B around the top until it meets point A. The line is now a tangent to the circle, and PA=PB. Since PA=PB, then their product is equal to PA2. So:

PA2 = PC.PD

This is the Tangent-Secant Theorem. If you move point B around until it overlaps A, the resulting tangent has a length equal to PA2. Similarly, if you drag D around the bottom to point C, the that tangent has a length of PC2. From the this theorem

PA2 = PC2

By taking the square root of each side:

PA = PC

confirming that the two tangents froma point to a circle are always equal.

Inscribed angle
Central angle
Central angle theorem

This is the idea (a,b and c are angles):

And here it is with some actual values:

In words: the angle made by two secants (a line that cuts a circle at two points) that intersect outside the circle is half of the furthest arc minus the nearest arc.

Why not try drawing one yourself, measure it using a protractor,
and see what you get?

It also works when either line is a tangent (a line that just touches a circle at one point). Here we see the "both are tangents" case:

That's it! You know it now.

Is this magic?

Well, we can prove it if you want:

AC and BD are two secants that intersect at the point P outside the circle. What is the relationship between the angle CPD and the arcs AB and CD?

We start by saying that the angle subtended by arc CD at O is 2θ and the arc subtended by arc AB at O is 2Φ

By the Angle at the Center Theorem:

∠DAC = ∠DBC = θ and ∠ADB = ∠ACB = Φ

And PAC is 180°, so:

∠DAP = 180° − θ

Now use angles of a triangle add to 180° in triangle APD:

∠CPD = 180° − (∠DAP + ∠ADP)

∠CPD = 180° − (180° − θ + Φ) = θ − Φ

∠CPD = θ − Φ

∠CPD = ½(2θ − 2Φ)

Done!

This is the idea (a,b,c and d are lengths):

And here it is with some actual values (measured only to whole numbers):

And we get

12 × 25 = 300
13 × 23 = 299

Very close! If we measured perfectly the results would be equal.

Why not try drawing one yourself, measure the lengths and see what you get?

The lines are called secants (a line that cuts a circle at two points).

This also works if one or both are tangents (a line that just touches a circle at one point), but since two lengths are identical we don't write c×d or c×c we just write c2:

(Question: What happens when both are tangents?)

Because there are similar triangles! Looking below:

They both share the angle θ
They both have the same angle φ (see inscribed angles)

The triangles may not be the same size, but they have the same angles ... so all lengths will be in proportion!

Looking at the lengths coming from point "P", one triangle has the ratio a/d, and the other has the matching ratio c/b:

a/d = c/b

a × b = c × d

A line that intersects a circle in exactly one point is called a tangent and the point where the intersection occurs is called the point of tangency. The tangent is always perpendicular to the radius drawn to the point of tangency.

A secant is a line that intersects a circle in exactly two points.

When a tangent and a secant, two secants, or two tangents intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

$$m\angle A=\frac{1}{2}(m\overline{DE}-m\overline{BC} )$$

When two chords intersect inside a circle, then the measures of the segments of each chord multiplied with each other is equal to the product from the other chord:

$$AB\cdot EB=CE\cdot ED$$

If two secants are drawn to a circle from one exterior point, then the product of the external segment and the total length of each secant are equal:

$$AB\cdot AD=AC\cdot AE$$

If one secant and one tangent are drawn to a circle from one exterior point, then the square of the length of the tangent is equal to the product of the external secant segment and the total length of the secant:

$$AB^{2}=AC\cdot AD$$

If we have a circle drawn in a coordinate plane, with the center in (a,b) and the radius r then we could always describe the circle with the following equation:

$$(x-a)^{2}+(y-b)^{2}=r^{2}$$

Video lesson

Find the value of t in the figure

Prove and use theorems involving lines that intersect a circle at two points.

When two secants intersect outside a circle, the circle divides the secants into segments that are proportional with each other.

Two Secants Segments Theorem: If two secants are drawn from a common point outside a circle and the segments are labeled as below, then $a(a+b)=c(c+d)$.

Figure $\PageIndex{1}$

What if you were given a circle with two secants that intersect outside the circle? How could you use the length of some of the segments formed by their intersection to determine the lengths of the unknown segments?

Example $\PageIndex{1}$

Find $x$. Simplify any radicals.

Figure $\PageIndex{2}$

Solution

Use the Two Secants Segments Theorem.

$\begin{aligned} 8(8+x)&=6(6+18) \\ 64+8x&=144 \\ 8x&=80 \\ x&=10\end{aligned}$

Example $\PageIndex{2}$

Find $x$. Simplify any radicals.

Figure $\PageIndex{3}$

Solution

Use the Two Secants Segments Theorem.

$\begin{aligned} 15(15+27)&=x\cdot45 \\ 630&=45x \\ x&=14 \end{aligned}$

Example $\PageIndex{3}$

Find the value of $x$.

Figure $\PageIndex{4}$

Solution

Use the Two Secants Segments Theorem.

$\begin{aligned}18\cdot(18+x)&=16\cdot(16+24) \\ 324+18x&=256+384 \\ 18x&=316 \\ x&=17\dfrac{5}{9}\end{aligned}$

Example $\PageIndex{4}$

Find the value of $x$.

Figure $\PageIndex{5}$

Solution

Use the Two Secants Segments Theorem.

$\begin{aligned}x\cdot(x+x)&=9\cdot 32 \\ 2x^2&=288 \\ x^2&=144 \\ x&=12,\: x\neq −12 (\text{length is not negative})\end{aligned}$

Example $\PageIndex{5}$

True or False: Two secants will always intersect outside of a circle.

Solution

False. If the two secants are parallel, they will never intersect. It's also possible for two secants to intersect inside a circle.

Fill in the blanks for each problem below. Then, solve for the missing segment.

Figure $\PageIndex{6}$

$3(\text{______}+\text{______})=2(2+7)$

Figure $\PageIndex{7}$

$x\cdot\text{______}=8(\text{______}+\text{______})$

Find x in each diagram below. Simplify any radicals.

Figure $\PageIndex{8}$
Figure $\PageIndex{9}$
Figure $\PageIndex{10}$

Fill in the blanks of the proof of the Two Secants Segments Theorem.
Figure $\PageIndex{11}$

Given: Secants $\overline{PR}$ and $\overline{RT}$

Prove: $a(a+b)=c(c+d)$

Statement	Reason
1. Secants $\overline{PR}$ and $\overline{RT}$ with segments $a$, $b$, $c$, and $d$.	1. Given
2. $\angle R\cong \angle R$	2. Reflexive PoC
3. $\angle QPS\cong \angle STQ$	3. Congruent Inscribed Angles Theorem
4. $\Delta RPS\sim \Delta RTQ$	4. AA Similarity Postulate
5. $ac+d=ca+b$	5. Corresponding parts of similar triangles are proportional
6. $a(a+b)=c(c+d)$	6. Cross multiplication

Solve for the unknown variable.

Figure $\PageIndex{12}$
Figure $\PageIndex{13}$
Figure $\PageIndex{14}$
Figure $\PageIndex{15}$
Figure $\PageIndex{16}$
Figure $\PageIndex{17}$
Figure $\PageIndex{18}$
Figure $\PageIndex{19}$
Figure $\PageIndex{12}$

To see the Review answers, open this PDF file and look for section 9.10.

Term	Definition
central angle	An angle formed by two radii and whose vertex is at the center of the circle.
chord	A line segment whose endpoints are on a circle.
circle	The set of all points that are the same distance away from a specific point, called the center.
diameter	A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Inscribed Angle	An inscribed angle is an angle with its vertex on the circle. The measure of an inscribed angle is half the measure of its intercepted arc.
intercepted arc	The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency	The point where the tangent line touches the circle.
radius	The distance from the center to the outer rim of a circle.
AA Similarity Postulate	If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
Congruent	Congruent figures are identical in size, shape and measure.
Reflexive Property of Congruence	$\overline{AB}\cong \overline{AB}$ or $\angle B\cong \angle B$
Secant	The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
secant line	A secant line is a line that joins two points on a curve.
Tangent line	A tangent line is a line that "just touches" a curve at a single point and no others.
Two Secants Segments Theorem	Two secants segments theorem states that if you have a point outside a circle and draw two secant lines from it, there is a relationship between the line segments formed.